As for the last counterterm, such a choice is unnatural and would be prohibited if the quantum theory of the vector field \(g_\mu \)  will be renormalizable (cf. the Coleman-Weinberg potential [1]} containing the term \(\phi ^4\ln \phi ^2\) ). Nevertheless, if we add such counterterms, the global conformal invariance of the renormalized effective action is broken as long as this logarithmic counterterm is not conformally invariant. The conformal anomaly is given by the factor at the logarithm. On the other hand, the second relation in (REF ) holds on a quantum level provided the power-like divergencies are completely canceled out. The finite part and the coefficient of the logarithm are invariant under the stretching of the Killing vector.
